TEKS Clarification Document Mathematics Algebra 2 2012 2013
111.31. Implementation of Texas Essential Knowledge and Skills for Mathematics, Grades 9-12. Source: The provisions of this 111.31 adopted to be effective September 1, 1996, 21 TexReg 7371; amended to be effective August 1, 2006, 30 TexReg 4479 111.33. Algebra II. (a) Basic understandings. (1) Foundation concepts for high school mathematics. As presented in Grades K-8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students continue to build on this foundation as they expand their understanding through other mathematical experiences. (2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students study algebraic concepts and the relationships among them to better understand the structure of algebra. (3) Functions, equations, and their relationship. The study of functions, equations, and their relationship is central to all of mathematics. Students perceive functions and equations as means for analyzing and understanding a broad variety of relationships and as a useful tool for expressing generalizations. (4) Relationship between algebra and geometry. Equations and functions are algebraic tools that can be used to represent geometric curves and figures; similarly, geometric figures can illustrate algebraic relationships. Students perceive the connections between algebra and geometry and use the tools of one to help solve problems in the other. (5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems. (6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, language and communication, and reasoning (justification and proof) to make connections within and outside mathematics. Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts. 2012, TESCCC 03/29/12 Page 2 of 66
2A.1 Foundations for functions. The student uses properties and attributes of functions and applies functions to problem situations. The student is expected to: 2A.1A Identify the mathematical domains and ranges of functions and determine reasonable domain and range values for continuous and discrete situations. Readiness Standard Use, Apply, Identify, Determine REASONABLE MATHEMATICAL DOMAINS AND RANGES Comparison of discrete and continuous domain Representations of domain and range with and without technology Tables Graphs Verbal descriptions Algebraic generalizations (including equation and function notation) Connections among the representations Notation of domain and range including set builder notation Lists of domain and range in sets (including f(x) for y) Verbal Ex: Domain is all real numbers; domain is all real numbers greater than five. Ex: Range is all real numbers less than zero; range is all real numbers. Symbolic Ex: domain (e.g., {x x > 0}; {x -3 < x < 4}; {x x R}; Ø) Ex: range (e.g., {y y 0}; {y -7 < y 4}; { f(x) -5 < f(x) 2}; {y y R}; Ø) Determination of scales for graphs and windows on graphing calculators using domain and range Contextual domain and range of real-world problem situation 2012, TESCCC 03/29/12 Page 3 of 66
Specific range values at given domain values from various representations (e.g., tables, graphs, or algebraic generalizations) Specific domain values at given range values from various representations (e.g., tables, graphs, or algebraic generalizations) Domain and range of the problem situation versus the domain and range of the mathematical function Note: In Algebra 1, students determine domain and range using various representations of functions. 2A.1B VII. Functions B1 Understand and analyze features of a function. Collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments. Readiness Standard Collect, Organize DATA Data sets and real-world problem situations Data collection and analysis with and without the use of technology (e.g., CBR/CBL, graphing calculators, and computers) Ex: measurement activities, experiments, simulations, Internet, etc. Representations of data with and without technology Tables Graphs including scatterplots Verbal descriptions Algebraic generalizations (including equation and function notation) Use, Make, Interpret, Fit, Model, Apply SCATTERPLOTS AND REPRESENTATIVE FUNCTIONS FOR DATA 2012, TESCCC 03/29/12 Page 4 of 66
Data sets and real-world problem situations General trends in data Data collection and analysis with and without the use of technology (e.g., CBR/CBL, graphing calculators, and computers) Ex: measurement activities, experiments, simulations, Internet, etc. Representations of data with and without technology Tables Graphs, including scatterplots Verbal descriptions Algebraic generalizations (including equation and function notation) Parent functions as representations of scatterplots of data with and without technology Linear Quadratic Square root Rational Exponential Logarithmic Absolute value Linear or nonlinear scatterplots using graphing technology Positive linear correlation Negative linear correlation No linear correlation Model, Predict, Make DECISIONS AND CRITICAL JUDGEMENTS IN PROBLEM SITUATIONS 2012, TESCCC 03/29/12 Page 5 of 66
Data sets and real-world problem situations General trends in the data with and without technology Data collection and analysis with and without the use of technology (e.g., CBR/CBL, graphing calculators, and computers) Ex: measurement activities, experiments, simulations, Internet, etc. Representations of data with and without technology Tables Graphs including scatterplots Verbal descriptions Algebraic generalizations (including equation and function notation) Parent functions as representations of scatterplots of data with and without technology Linear Quadratic Square root Rational Exponential Logarithmic Absolute value Linear or nonlinear scatterplots using graphing technology Positive linear correlation Negative linear correlation No linear correlation Predictions and critical judgments in terms of the scatterplots and representative functions Note: In Algebra 1, students collect and analyze data using various representations, which are then used to make predictions and critical judgments. 2012, TESCCC 03/29/12 Page 6 of 66
II. Algebraic Reasoning D1 Interpret multiple representations of equations and relationships. II. Algebraic Reasoning D2 Translate among multiple representations of equations and relationships. III. Geometric Reasoning C2 Make connections between geometry, statistics, and probability. IV. Measurement Reasoning D1 Compute and use measures of center and spread to describe data. VI. Statistical Reasoning A1 Plan a study. VI. Statistical Reasoning B1 Determine types of data. VI. Statistical Reasoning B2 Select and apply appropriate visual representations of data. VI. Statistical Reasoning B4 Describe patterns and departure from patterns in a set of data. VI. Statistical Reasoning C1 Make predictions and draw inferences using summary statistics. VI. Statistical Reasoning C2 Analyze data sets using graphs and summary statistics. VI. Statistical Reasoning C3 Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software. VII. Functions A2 Recognize and distinguish between different types of functions. VII. Functions B2 Algebraically construct and analyze new functions. VII. Functions C1 Apply known function models. VII. Functions C2 Develop a function to model a situation. VIII. Problem Solving and Reasoning B2 Use various types of reasoning. IX. Communication and Representation B1 Model and interpret mathematical ideas and concepts using multiple representations. IX. Communication and Representation B2 Summarize and interpret mathematical information provided orally, visually, or in written form within the given context. IX. Communication and Representation C1 Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, graphs, and words. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. 2A.2 Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to: 2A.2A Use tools including factoring and properties of exponents to simplify expressions and to transform and solve equations. Supporting Standard Use MATHEMATICAL TOOLS 2012, TESCCC 03/29/12 Page 7 of 66
Factoring Greatest common factor Difference of squares Sum and difference of cubes Trinomials Algebraic properties of equations and inequalities Ex: commutative, associative, distributive, identities, inverses, etc. Properties of exponents Zero power Negative exponents Multiplication law of exponents Division law of exponents Power law of exponents Rational exponents Manipulate, Simplify EXPRESSIONS Mathematical expressions Evaluation of expressions when given values for variables Simplification of expressions using properties of algebra Polynomial expressions Characteristics of polynomials Classifications of polynomials Operations with polynomials Factoring 2012, TESCCC 03/29/12 Page 8 of 66
Algebraic generalization of a data set or problem situation Problem situations involving simplifying polynomials and operations with polynomials Functional expressions Equation notation Ex: If (2, y) is a solution to the equation, find y if 2x 3y = 7. Ex: Given the equation y = 2x + 3, where y is dependent on x, what are the dependent values if x {-4, 0, 2, 12}? Function notation Ex: If f(x) = 2x 1, find f(2). Ex: If f(x) = 2x 1 and f(x) = 5, find x. Function values using technology (e.g., the table or trace features on the graphing calculator) Tools (e.g., exponent rules, factoring, algebraic properties) Manipulate, Transform, Solve EQUATIONS AND INEQUALITIES Distinction between expressions and equations and the difference between simplifying and solving (TxCCRS) Types of equations and inequalities Linear Quadratic Square root Rational Exponential Logarithmic Tools (e.g., factoring, algebraic properties, and rules of exponents) Transformation of equations and inequalities using properties of equality and/or properties of inequalities 2012, TESCCC 03/29/12 Page 9 of 66
Note: In Algebra 1, students simplify expressions, apply rules of exponents, and solve linear and quadratic equations. 2A.2B II. Algebraic Reasoning A1 Explain and differentiate between expressions and equations using words such as solve, evaluate, and simplify. II. Algebraic Reasoning B1 Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions). II. Algebraic Reasoning C1 Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations. Use complex numbers to describe the solutions of quadratic equations. Supporting Standard Use, Describe COMPLEX NUMBERS AS SOLUTIONS TO QUADRATIC EQUATIONS Complex number system Definitions and characteristics of real and imaginary parts Complex numbers on the graphing calculator Methods of determining complex solutions Graph of function (indicative of complex solution, does not cross x-axis) Algebraic manipulation Quadratic formula Note: In Algebra 1, students solve quadratic equations for real solutions only. I. Numeric Reasoning A2 Define and give examples of complex numbers. 2012, TESCCC 03/29/12 Page 10 of 66
2A.3 Foundations for functions. The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. The student is expected to: 2A.3A Analyze situations and formulate systems of equations in two or more unknowns or inequalities in two unknowns to solve problems. Readiness Standard Formulate, Use, Solve, Analyze SYSTEMS OF EQUATIONS AND INEQUALITIES FOR PROBLEM SITUATIONS Systems of equations Two or more unknowns Two or more equations Linear and nonlinear Systems of inequalities Two unknowns Two inequalities Linear and nonlinear Problem situations represented by systems of equations Problem situations represented by systems of inequalities Linear programming problems Note: In Algebra 1, students formulate 2 x 2 systems of equations, solve 2 x 2 systems using various methods, and justify the reasonableness of the solutions. Algebra 2 is the first time systems of inequalities are introduced, and it is not limited to linear systems. VII. Functions C2 Develop a function to model a situation. IX. Communication and Representation B2 Summarize and interpret mathematical information provided orally, visually, or in written form within the 2012, TESCCC 03/29/12 Page 11 of 66
2A.3B given context. Use algebraic methods, (i.e., substitution and linear combinations) graphs, tables, or matrices, to solve systems of equations or inequalities. Readiness Standard Use, Solve SYSTEMS OF EQUATIONS AND INEQUALITIES Distinction between expressions and equations and the difference between simplifying and solving (TxCCRS) Methods for solving systems of equations with and without technology Tables Common points on tables Graphs Identification of possible solutions in terms of points of intersection Algebraic methods Substitution Linear combination (elimination) Matrices (only technology) Method for solving system of inequalities with and without technology Graphical analysis of system Graphing of each function Shading of inequality region for each function Identification of common region of intersection Methods for solving linear programming problems with and without technology Graphical analysis of system Graphing of each function 2012, TESCCC 03/29/12 Page 12 of 66
Shading of inequality region for each function Identification of common region of intersection Determination of points of intersection by solving system of equations Conclusion in terms of the problem situation Comparison of solutions to equations and inequalities in a problem situation (TxCCRS) Relationships and connections between the methods of solution Representation of the solution as a point of intersection (system of equations) Representation of the solution as a point in the solution region (system of inequalities) Interpretation of a solution point in terms of the problem situation Note: In Algebra 1, students formulate 2 x 2 systems of linear equations, solve 2 x 2 systems using various methods, and justify the reasonableness of the solutions. 2A.3C II. Algebraic Reasoning A1 Explain and differentiate between expressions and equations using words such as solve, evaluate, and simplify. II. Algebraic Reasoning C2 Explain the difference between the solution set of an equation and the solution set of an inequality. II. Algebraic Reasoning C1 Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations. IX. Communication and Representation B1 Model and interpret mathematical ideas and concepts using multiple representations. Interpret and determine the reasonableness of solutions to systems of equations or inequalities for given contexts. Readiness Standard Analyze, Interpret, Determine REASONABLENESS OF SOLUTIONS TO SYSTEMS OF EQUATIONS AND INEQUALITIES Justification of solutions to systems of equations with and without technology Verbal description 2012, TESCCC 03/29/12 Page 13 of 66
Tables Graphs Substitution of solutions into original functions Justification of solutions to system of inequalities with and without technology Verbal description Tables Graphs Substitution of solutions into original functions Justification of reasonableness of solution in terms of the problem situation or data collection Note: In Algebra 1, students formulate 2 x 2 systems of linear equations, solve 2 x 2 systems using various methods, and justify the reasonableness of the solutions. II. Algebraic Reasoning C2 Explain the difference between the solution set of an equation and the solution set of an inequality. II. Algebraic Reasoning D1 Interpret multiple representations of equations and relationships. 2A.4 Algebra and geometry. The student connects algebraic and geometric representations of functions. The student is expected to: 2A.4A Identify and sketch graphs of parent functions, including linear (f(x) = x), quadratic (f(x) = x²), exponential (f(x) = a x ), and logarithmic (f(x) = log a x) functions, absolute value of x (f(x) = x ), square root of x (f(x) = x ), and reciprocal of x (f(x) = 1 x ). Supporting Standard Connect, Identify, Sketch PARENT FUNCTIONS AND THEIR GRAPHS General forms of parent functions (including equation and function notation) Linear: f(x) = x 2012, TESCCC 03/29/12 Page 14 of 66
Quadratic: f(x) = x² Exponential: f(x) = a x Logarithmic: f(x) = log a x Absolute value: f(x) = x Square root: f(x) = x Rational (reciprocal of x): f(x) = 1 x Representations of parent functions with and without technology Tables Graphs Verbal descriptions Algebraic generalizations (including equation and function notation) Connections between representations Characteristics of parent functions with and without technology Ex: domain and range, rate of change, increasing/decreasing, vertex points, end points, intercepts, symmetry, asymptotes, maximum/minimum, end behavior, etc. Connections between transformed functions to the respective parent function with and without technology Similarities and differences in the parent functions Identification of relations as functions (TxCCRS) Note: In Algebra 1, students identified the characteristics of the linear and quadratic parent functions. III. Geometric Reasoning C1 Make connections between geometry and algebra. VII. Functions A1 Recognize whether a relation is a function. VII. Functions A2 Recognize and distinguish between different types of functions. VII. Functions B1 Understand and analyze features of a function. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. 2012, TESCCC 03/29/12 Page 15 of 66
2A.4B Extend parent functions with parameters such as a in f(x) = a/x and describe the effects of the parameter changes on the graph of parent functions. Readiness Standard Extend, Describe PARAMETER CHANGES ON PARENT FUNCTIONS General forms of parent functions (including equation and function notation) Linear: f(x) = x Quadratic: f(x) = x² Exponential: f(x) = b x Logarithmic: f(x) = log b x Absolute value: f(x) = x Square root: f(x) = x Rational (reciprocal of x): f(x) = 1 x Representations of parent functions with and without technology Graphs Verbal descriptions Algebraic generalizations (including equation and function notation) a Changes in parameters a, h, and k on the graph of the parent function: y = k x h + or Effects of changes in a on the graph of the parent function with and without technology a 0 a > 1, the graph stretches vertically 0 < a < 1, the graph compresses vertically 1 y = a + k x h 2012, TESCCC 03/29/12 Page 16 of 66
Opposite of a reflects vertically over the horizontal axis (x-axis) Effects of changes in h on the graph of the parent function with and without technology h = 0, no horizontal shift Horizontal shift left or right Effects of changes in k on the graph of the parent function with and without technology k = 0, no vertical shift Vertical shift up or down Determination of parameter changes given a graphical or algebraic representation Determination of a graphical representation given the algebraic representation or parameter changes Determination of an algebraic representation given the graphical representation or parameter changes Connections between the critical attributes of transformed functions and their respective parent functions Descriptions of the effects on the domain and range by the parameter changes Effects of parameter changes in mathematical situations Effects of parameter changes in problem situations Note: In Algebra 1, students extended parameters on vertical shifts, stretches, and compressions for linear and quadratic functions. In the exponential and logarithmic parent functions, the a, used as the exponent and log base, was changed to b, so it would not be confused with the a parameter used in transformations of parent functions. 2A.4C III. Geometric Reasoning B1 Identify and apply transformations to figures. III. Geometric Reasoning C1 Make connections between geometry and algebra. VII. Functions B1 Understand and analyze features of a function. VII. Functions B2 Algebraically construct and analyze new functions. VII. Functions C1 Apply known function models. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. Describe and analyze the relationship between a function and its inverse. Supporting Standard 2012, TESCCC 03/29/12 Page 17 of 66
Describe, Analyze RELATIONSHIPS BETWEEN FUNCTIONS AND THEIR INVERSES Comparison of independent and dependent variables of a function and its inverse Comparison of domain and range of a function and its inverse Symmetry about y = x Identification of functions and non-functions Comparison of functions and their inverses using multiple representations with and without technology Tables Graphs Verbal descriptions Algebraic generalizations (including equation and function notation) Inverse relationships Quadratic and square root functions Exponential and logarithmic functions III. Geometric Reasoning C1 Make connections between geometry and algebra. VII. Functions A1 Recognize whether a relation is a function. VII. Functions B1 Understand and analyze features of a function. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. 2A.5 Algebra and geometry. The student knows the relationship between the geometric and algebraic descriptions of conic sections. The student is expected to: 2A.5A Describe a conic section as the intersection of a plane and a cone. Supporting Standard Describe 2012, TESCCC 03/29/12 Page 18 of 66
CONIC SECTIONS AS INTERSECTIONS OF A PLANE AND A CONE Hands-on activities to model the intersection of a plane and a cone Geometric and algebraic descriptions of the intersection Circle Ellipse Parabola Hyperbola Characteristics of conic sections Center Vertices Symmetry Real-world examples of conic sections 2A.5B III. Geometric Reasoning A1 Identify and represent the features of plane and space figures. III. Geometric Reasoning C1 Make connections between geometry and algebra. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. Sketch graphs of conic sections to relate simple parameter changes in the equation to corresponding changes in the graph. Supporting Standard Sketch, Relate EFFECTS OF PARAMETER CHANGES IN EQUATIONS TO GRAPHS OF CONIC SECTIONS Analysis of graphs of conic sections with and without technology 2012, TESCCC 03/29/12 Page 19 of 66
General form: Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 Changes in parameters from working or standard form Circle: a, b, h, and k on the graph 2 2 2 ( x h) + ( y k) = r, where r is the radius of the circle 2 2 ( x h) ( y k) + = 1, where a = b 2 2 a b Ellipse: a, b, h, and k on the graph MATHEMATICS TEKS CLARIFICATION DOCUMENT 2 2 2 2 ( x h) ( y k) ( y k) ( x h) + = 1 or + = 1, where a b 2 2 2 2 a b a b 2 2 ( x h) ( y k) + = 1, where a b 2 2 a b Hyperbola: a, b, h, and k on the graph 2 2 2 2 ( x h) ( y k) ( y k) ( x h) = 1 or = 1 2 2 2 2 a b a b 2 2 ( x h) ( y k) ± = 1 2 2 a b Parabola: a, p, h, and k on the graph 2 2 ( x h) = 4p( y k) or ( y k) = 4p( x h) y = a( x h) 2 + k or ( ) 2 x = a y h + k, where a = 1 4p Completing the square to transform the general form of an equation representing a conic section to standard form of the equation Effects of parameter change on location of the center, vertices, line(s) of symmetry, directrix, asymptotes, and focal points Effects of parameter change on the domain and range of the transformed graph Graph of a transformed conic section given the standard form of the equation Standard form of an equation given the graph of a transformed conic section 2012, TESCCC 03/29/12 Page 20 of 66
2A.5C II. Algebraic Reasoning D2 Translate among multiple representations of equations and relationships. III. Geometric Reasoning A1 Identify and represent the features of plane and space figures. III. Geometric Reasoning B1 Identify and apply transformations to figures. III. Geometric Reasoning B3 Use congruence transformations and dilations to investigate congruence, similarity, and symmetries of plane figures. III. Geometric Reasoning C1 Make connections between geometry and algebra. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. Identify symmetries from graphs of conic sections. Supporting Standard Know, Identify SYMMETRY OF CONIC SECTIONS Analysis of graphs of conic sections with and without technology Symmetrical aspects on the graphs of conic sections Symmetries around the vertical and horizontal axes of the conic sections Equations of the lines of symmetry from the graphs Effects of parameter changes on lines of symmetry 2A.5D III. Geometric Reasoning B2 Identify symmetries of a plane figure. III. Geometric Reasoning B3 Use congruence transformations and dilations to investigate congruence, similarity, and symmetries of plane figures. III. Geometric Reasoning C1 Make connections between geometry and algebra. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. Identify the conic section from a given equation. Supporting Standard Know, Identify 2012, TESCCC 03/29/12 Page 21 of 66
CONIC SECTIONS FROM EQUATIONS General form: Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 Comparisons by coefficients of the squared terms when B= 0 Circle: A = C Ellipse: A C but same sign Hyperbola: A and C have different sign Parabola: Either A = 0 or C = 0, but not both Conic section by discriminant: b 2 4ac Discriminant < 0: ellipse if A C or circle if A = C Discriminant = 0: parabola Discriminant > 0: hyperbola Working or standard form Circle 2 2 2 ( x h) + ( y k) = r, where r is the radius of the circle 2 2 ( x h) ( y k) + = 1, where a = b 2 2 a b Ellipse 2 2 2 2 ( x h) ( y k) ( y k) ( x h) + = 1 or + = 1, where a b 2 2 2 2 a b a b 2 2 ( x h) ( y k) + = 1, where a b 2 2 a b Hyperbola 2 2 2 2 ( x h) ( y k) ( y k) ( x h) = 1 or = 1 2 2 2 2 a b a b 2012, TESCCC 03/29/12 Page 22 of 66
2 2 ( x h) ( y k) ± = 1 2 2 a b Parabola 2 2 ( x h) = 4p( y k) or ( y k) = 4p( x h) y = a( x h) 2 + k or ( ) 2 x = a y h + k, where a = General form of the circle, ellipse, hyperbola equation Standard form of the circle, ellipse, hyperbola equation General and vertex form of a parabola 1 4p 2A.5E II. Algebraic Reasoning D2 Translate among multiple representations of equations and relationships. III. Geometric Reasoning C1 Make connections between geometry and algebra. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. Use the method of completing the square. Supporting Standard Know, Use METHOD OF COMPLETING THE SQUARE Concrete models (e.g., algebra tiles) and pictorial representations to complete the square Completing the square to transform the general form of an equation representing a conic section to standard form of the equation III. Geometric Reasoning C1 Make connections between geometry and algebra. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. 2A.6 Quadratic and square root functions. The student understands that quadratic functions can be represented in different ways and 2012, TESCCC 03/29/12 Page 23 of 66
2A.6A translates among their various representations. The student is expected to: Determine the reasonable domain and range values of quadratic functions, as well as interpret and determine the reasonableness of solutions to quadratic equations and inequalities. Readiness Standard Understand, Represent, Determine, Interpret DOMAIN AND RANGE OF QUADRATIC FUNCTIONS Data collections and problem situations involving quadratic relationships Comparison of discrete and continuous domain Representations of domain and range with and without technology Tables Graphs Verbal descriptions Algebraic generalizations (including equation and function notation) Characteristics of quadratic equations/inequalities related to determining domain and range with and without technology Vertex Maximum Minimum Intercepts Axis of symmetry End behaviors Notation of domain and range including set builder notation Lists of domain and range in sets (including f(x) for y) Verbal Ex: Domain is all real numbers; domain is all real numbers greater than five. 2012, TESCCC 03/29/12 Page 24 of 66
Ex: Range is all real numbers less than zero; range is all real numbers. Symbolic Ex: domain (e.g., {x x > 0}; {x -3 < x < 4}; {x x R}; Ø) Ex: range (e.g., {y y 0}; {y -7 < y 4}; { f(x) -5 < f(x) 2}; {y y R}; Ø) Determination of scales for graphs and windows on graphing calculators using domain and range Specific range values at given domain values from various representations (e.g., tables, graphs, or algebraic generalizations) Specific domain values at given range values from various representations (e.g., tables, graphs, or algebraic generalizations) Domain and range of the problem situation versus the domain and range of the mathematical function Understand, Determine, Interpret REASONABLENESS OF SOLUTIONS TO QUADRATIC EQUATIONS AND INEQUALITIES Justification of solutions to quadratic equations with and without technology Verbal Tabular Graphical Algebraic by substitution of solutions into original function Justification of solutions to quadratic inequalities with and without technology Verbal Tabular Graphical Algebraic by substitution of solutions into original function Justification of solutions in a problem situation involving quadratic relationships Differentiation between the domain and range of the function and the reasonable domain and range for the real-world situation Note: In Algebra 1, students determined the domain and range of quadratic functions. 2012, TESCCC 03/29/12 Page 25 of 66
2A.6B I. Numeric Reasoning C1 Use estimation to check for errors and reasonableness of solutions. II. Algebraic Reasoning D1 Interpret multiple representations of equations and relationships. VII. Functions B1 Understand and analyze features of a function. VIII. Problem Solving and Reasoning C2 Use a function to model a real world situation. Relate representations of quadratic functions, such as algebraic, tabular (chart), graphical, and verbal descriptions to one another. Readiness Standard Relate, Translate REPRESENTATIONS OF QUADRATIC FUNCTIONS Representations of quadratic functions with and without technology Graphs (graphical) Tables (tabular (chart)) Verbal description Algebraic generalization (including equation and function notation) Characteristics of quadratic functions with and without technology Domain Range Maximum Minimum Vertex Intercepts Zeros Axis of symmetry Symmetric points 2012, TESCCC 03/29/12 Page 26 of 66
End behaviors Connections between characteristics in the various representations Note: In Algebra 1, students represent quadratic functions using tables, graphs, verbal descriptions, and algebraic generalizations. 2A.6C II. Algebraic Reasoning D2 Translate among multiple representations of equations and relationships. VII. Functions B1 Understand and analyze features of a function. VII. Functions B2 Algebraically construct and analyze new functions. VII. Functions C2 Develop a function to model a situation. IX. Communication and Representation A3 Use mathematics as a language for reasoning, problem solving, making connections, and generalizing. IX. Communication and Representation B1 Model and interpret mathematical ideas and concepts using multiple representations. IX. Communication and Representation B2 Summarize and interpret mathematical information provided orally, visually, or in written form within the given context. IX. Communication and Representation C1 Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, graphs, and words. Determine a quadratic function from its roots (real and complex) or a graph. Supporting Standard Represent, Determine, Understand QUADRATIC FUNCTIONS Determination of a quadratic function with and without technology Set of any three points on the graph using matrices and/or regression Graph of a quadratic function Zeros x-intercepts Roots of an equation (real and complex) 2012, TESCCC 03/29/12 Page 27 of 66
Real roots including operations with radicals (TxCCRS) Complex roots including operations with complex numbers (TxCCRS) Association of solutions and roots to quadratic equations Association of zeros and horizontal intercepts (x-intercepts) to graphs of functions Connections between solutions and roots of quadratic equations to the zeros and x-intercepts of the related function Quadratic model from a scatterplot using a graphing calculator Note: In Algebra 1, students made connections among solutions (roots) of quadratic equations and zeros (x-intercepts) of quadratic functions. I. Numeric Reasoning B1 Perform computations with real and complex numbers. VII. Functions B2 Algebraically construct and analyze new functions. VII. Functions C2 Develop a function to model a situation. 2A.7 Quadratic and square root functions. The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations. The student is expected to: 2A.7A Use characteristics of the quadratic parent function to sketch the related graphs and connect between the y = ax 2 + bx + c and the y = a(x - h) 2 + k symbolic representations of quadratic functions. Readiness Standard Interpret, Describe, Use, Sketch CHARACTERISTICS OF THE QUADRATIC PARENT FUNCTION FOR GRAPHING RELATED FUNCTIONS Comparison of quadratic parent function y = x 2 to transformations reflected in y = a(x - h) 2 + k with and without technology Representations of parent function and related function with and without technology Tables Graphs Verbal descriptions 2012, TESCCC 03/29/12 Page 28 of 66
Algebraic generalizations (including equation and function notation) Changes in parameters a, h, and k on the graph of the parent function, y = a( x h) 2 + k Effects of changes in a on the graph of the parent function with and without technology a 0 a > 1, the graph stretches vertically 0 < a < 1, the graph compresses vertically Opposite of a reflects vertically over the horizontal axis (x-axis) Effects of changes in h on the graph of the parent function with and without technology h = 0, no horizontal shift Horizontal shift left or right Effects of changes in k on the graph of the parent function with and without technology k = 0, no vertical shift Vertical shift up or down Determination of parameter changes given a graphical or algebraic representation Determination of a graphical representation given the algebraic representation or parameter changes Determination of an algebraic representation given the graphical representation or parameter changes Connections between the critical attributes of transformed functions and their respective parent functions Descriptions of the effects on the domain and range by the parameter changes Effects of parameter changes in mathematical situations Effects of parameter changes in problem situations Connect STANDARD AND VERTEX FORM OF QUADRATIC FUNCTIONS x-value of the vertex as h and the y-value of the vertex as k in y = a( x h) 2 + k 2012, TESCCC 03/29/12 Page 29 of 66
x-value of the vertex (h) in y = a( x h) 2 + k to c as the y-intercept in 2 y = ax + bx + c MATHEMATICS TEKS CLARIFICATION DOCUMENT b x = from 2a Completing the square to transform from the standard form 2 y = ax + bx + c 2 y ax bx c y = a x h + k = + + to vertex form ( ) 2 Squaring the binomial in y = a( x h) 2 + k and simplifying to find the standard form Characteristics of 2 y ax bx c Vertex (maximum or minimum) Axis of symmetry y-intercept Symmetric points = + + connected to characteristics of ( ) 2 2 y = ax + bx + c y = a x h + k on the graph with and without technology 2A.7B III. Geometric Reasoning B1 Identify and apply transformations to figures. III. Geometric Reasoning C1 Make connections between geometry and algebra. VII. Functions B1 Understand and analyze features of a function. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. Use the parent function to investigate, describe, and predict the effects of changes in a, h, and k on the graphs of y = a(x - h) 2 + k form of a function in applied and purely mathematical situations. Supporting Standard Interpret, Use, Investigate, Describe, Predict PARAMETER CHANGES ON QUADRATIC FUNCTIONS IN APPLIED AND MATHEMATICAL SITUATIONS Comparison of quadratic parent function y = x 2 to transformations reflected in y = a(x - h) 2 + k with and without technology Representations of parent function and related function with and without technology 2012, TESCCC 03/29/12 Page 30 of 66
Tables Graphs Verbal descriptions MATHEMATICS TEKS CLARIFICATION DOCUMENT Algebraic generalizations (including equation and function notation) Changes in parameters a, h, and k on the graph of the parent function, y = a( x h) 2 + k Effects of changes in a on the graph of the parent function with and without technology a 0 a > 1, the graph stretches vertically 0 < a < 1, the graph compresses vertically Opposite of a reflects vertically over the horizontal axis (x-axis) Effects of changes in h on the graph of the parent function with and without technology h = 0, no horizontal shift Horizontal shift left or right Effects of changes in k on the graph of the parent function with and without technology k = 0, no vertical shift Vertical shift up or down Determination of parameter changes given a graphical or algebraic representation Determination of a graphical representation given the algebraic representation or parameter changes Determination of an algebraic representation given the graphical representation or parameter changes Connections between the critical attributes of transformed functions and their respective parent functions Descriptions of the effects on the domain and range by the parameter changes Effects of parameter changes in mathematical situations Effects of parameter changes in problem situations Note: In Algebra 1, students investigate and describe the effects of changes in a and c on the graph of y = ax 2 + c. 2012, TESCCC 03/29/12 Page 31 of 66
III. Geometric Reasoning B1 Identify and apply transformations to figures. VII. Functions B1 Understand and analyze features of a function. 2A.8 Quadratic and square root functions. The student formulates equations and inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to: 2A.8A Analyze situations involving quadratic functions and formulate quadratic equations or inequalities to solve problems. Readiness Standard Formulate, Use, Solve QUADRATIC EQUATIONS AND INEQUALITIES FOR PROBLEM SITUATIONS Data collection activities with and without technology Real-world problem situations Determination of a quadratic equation by tables, graphs, verbal descriptions, technology (graphing calculator) Methods for solving quadratic equations with and without technology Tables Graphs Algebraic Factoring Algebraic manipulation Quadratic formula Solution sets of quadratic equations and inequalities Equations Two solutions Ex: x = -2 and 5; {-2, 5}; x = -2 and x = 5; (-2, 0) and (5, 0) One solution 2012, TESCCC 03/29/12 Page 32 of 66
Ex: -2; {-2}; x = -2; (-2, 0) which is a double root No solution Ex: no real roots, x = ± 3i ; { 3, i 3i }; x = 3i and x = 3i Inequalities One-dimensional Ex: -5 < x < 2 and a number line with open circles at -5 and 2 shaded in between Ex: x 2 or x -5 and a number line with closed circles at -5 and 2 shading out from each Two-dimensional Ex: y > 2x 2 + 3 or f(x) > 2x 2 + 3 and a dotted curve with shading above on a coordinate plane Ex: y 2x 2 + 3 or f(x) 2x 2 + 3 and a solid curve with shading below on a coordinate plane Comparison of solution sets of equations and inequalities (TxCCRS) Reasonableness of solutions mathematically and in context of problem situation 2A.8B II. Algebraic Reasoning C2 Explain the difference between the solution set of an equation and the solution set of an inequality. VII. Functions B2 Algebraically construct and analyze new functions. VII. Functions C1 Apply known function models. VII. Functions C2 Develop a function to model a situation. VIII. Problem Solving and Reasoning C2 Use a function to model a real world situation. Analyze and interpret the solutions of quadratic equations using discriminants and solve quadratic equations using the quadratic formula. Supporting Standard Analyze, Interpret SOLUTIONS OF QUADRATIC EQUATIONS Analysis of quadratic equations by discriminants 2 Substitution of values of a, b, and c into b 4ac 2012, TESCCC 03/29/12 Page 33 of 66
= 0, one rational double root > 0 and perfect square, two rational roots > 0 and not perfect square, two irrational roots (conjugates) < 0, two imaginary roots (conjugates) Solve QUADRATIC EQUATIONS Methods for solving quadratic equations with and without technology Factoring Completing the square 2 b ± b 4ac Quadratic formula: x = 2a Comparisons of methods for solving quadratic equations Types of solutions One rational double root Two rational roots Two irrational roots (conjugates) Two imaginary roots (conjugates) Note: In Algebra 1, students solved quadratic equations using concrete models, tables, graphs, factoring, and the quadratic formula. I. Numeric Reasoning A2 Define and give examples of complex numbers. II. Algebraic Reasoning A1 Explain and differentiate between expressions and equations using words such as solve, evaluate, and simplify. II. Algebraic Reasoning C1 Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations. 2012, TESCCC 03/29/12 Page 34 of 66
2A.8C II. Algebraic Reasoning D1 Interpret multiple representations of equations and relationships. Compare and translate between algebraic and graphical solutions of quadratic equations. Supporting Standard Analyze, Compare, Translate ALGEBRAIC AND GRAPHICAL SOLUTIONS OF QUADRATICS Algebraic solutions from quadratic equations Roots of the equation Factoring Quadratic formula Graphical solutions from quadratic functions with and without technology Zeros of the function x-intercepts Table comparisons Connections between solutions and roots of quadratic equations to the zeros and x-intercepts of the related function Connections of real and complex solutions to x-intercepts on the graph with and without technology Note: In Algebra 1, students make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function. 2A.8D II. Algebraic Reasoning D2 Translate among multiple representations of equations and relationships. Solve quadratic equations and inequalities using graphs, tables, and algebraic methods. Readiness Standard 2012, TESCCC 03/29/12 Page 35 of 66
Analyze, Solve QUADRATIC EQUATIONS AND INEQUALITIES Problem situations involving quadratic functions Distinction between expressions and equations and the difference between simplifying and solving (TxCCRS) Methods for solving quadratic equations with and without technology Tables Zeros Equal range values Graphs Zeros x-intercepts Algebraic methods Factoring Algebraic manipulation Quadratic formula Methods for solving quadratic inequalities with and without technology Graphs Tables Algebraic methods Factoring Algebraic manipulation Quadratic formula Testing and shading acceptable regions on a number line and coordinate plane Connections between solutions and roots of square root equations to the zeros and x-intercepts of the related function 2012, TESCCC 03/29/12 Page 36 of 66
Solution sets of quadratic equations and inequalities Equations Two solutions Ex: x = -2 and 5; {-2, 5}; x = -2 and x = 5; (-2, 0) and (5, 0) One solution Ex: -2; {-2}; x = -2; (-2, 0), which is a double root No solution Ex: no real roots Inequalities One-dimensional Ex: -5 < x < 2 and a number line with open circles at -5 and 2 shaded in between Ex: x 2 or x -5 and a number line with closed circles at -5 and 2 shading out from each Two-dimensional Ex: y > 2x 2 + 3 or f(x) > 2x 2 + 3 and a dotted curve with shading above on a coordinate plane Ex: y 2x 2 + 3 or f(x) 2x 2 + 3 and a solid curve with shading below on a coordinate plane Comparison of solution sets of equations and inequalities (TxCCRS) Comparison of solutions to equations and inequalities in a problem situation (TxCCRS) Reasonableness of solutions mathematically and in context of problem situation Note: In Algebra 1, students solved quadratic equations using concrete models, tables, graphs, factoring, and the quadratic formula. II. Algebraic Reasoning A1 Explain and differentiate between expressions and equations using words such as solve, evaluate, and simplify. II. Algebraic Reasoning C1 Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations. II. Algebraic Reasoning C2 Explain the difference between the solution set of an equation and the solution set of an inequality. 2A.9 Quadratic and square root functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to: 2012, TESCCC 03/29/12 Page 37 of 66
2A.9A Use the parent function to investigate, describe, and predict the effects of parameter changes on the graphs of square root functions and describe limitations on the domains and ranges. Supporting Standard Use, Investigate, Describe, Predict GRAPHS AND EFFECTS OF PARAMETER CHANGES ON SQUARE ROOT FUNCTIONS Comparison of square root parent function y Representations of parent function and related function with and without technology Tables Graphs Verbal descriptions Algebraic generalizations (including equation and function notation) Characteristics of the square root parent function with and without technology Endpoint Intercepts Symmetry Domain and range Single and compound inequality statements to identify domain and range Restriction on domain of square root function, x 0 End behavior = x to transformations reflected in y = a x h + k with and without technology Changes in parameters a, h, and k on the graph of the parent function, y = a x h + k Effects of changes in a on the graph of the parent function with and without technology a 0 a > 1, the graph stretches vertically 2012, TESCCC 03/29/12 Page 38 of 66
0 < a < 1, the graph compresses vertically Opposite of a reflects vertically over the horizontal axis (x-axis) Effects of changes in h on the graph of the parent function with and without technology h = 0, no horizontal shift Horizontal shift left or right Effects of changes in k on the graph of the parent function with and without technology k = 0, no vertical shift Vertical shift up or down Determination of parameter changes given a graphical or algebraic representation Determination of a graphical representation given the algebraic representation or parameter changes Determination of an algebraic representation given the graphical representation or parameter changes Connections between the critical attributes of transformed functions and their respective parent functions Descriptions of the effects on the domain and range and limitations of domain by the parameter changes Effects of parameter changes in mathematical situations Effects of parameter changes in problem situations 2A.9B III. Geometric Reasoning B1 Identify and apply transformations to figures. VII. Functions B1 Understand and analyze features of a function. VII. Functions B2 Algebraically construct and analyze new functions. Relate representations of square root functions, such as algebraic, tabular, graphical, and verbal descriptions with each other. Supporting Standard Relate, Analyze REPRESENTATIONS OF SQUARE ROOT FUNCTIONS Representations of square root functions with and without technology 2012, TESCCC 03/29/12 Page 39 of 66
Graphs (graphical) Tables (tabular) Verbal descriptions Algebraic generalizations (including equation and function notation) Characteristics of square root functions with and without technology Vertex point Intercepts Symmetry End behaviors Domain restrictions Connections between characteristics and representations with and without technology 2A.9C II. Algebraic Reasoning D2 Translate among multiple representations of equations and relationships. VII. Functions B1 Understand and analyze features of a function. VII. Functions C2 Develop a function to model a situation. IX. Communication and Representation B1 Model and interpret mathematical ideas and concepts using multiple representations. IX. Communication and Representation B2 Summarize and interpret mathematical information provided orally, visually, or in written form within the given context. Determine the reasonable domain and range values of square root functions, as well as interpret and determine the reasonableness of solutions to square root equations and inequalities. Supporting Standard Determine, Interpret, Analyze REASONABLE DOMAIN AND RANGE OF SQUARE ROOT FUNCTIONS Data collections and problem situations involving square root relationships Comparison of discrete and continuous domain 2012, TESCCC 03/29/12 Page 40 of 66